Abstract
In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let Un,g be the lollipop graph obtained by appending a pendent vertex of a path on n − g (n > g) vertices to a vertex of a cycle on g ⩾ 3 vertices. We prove that the graph Un,g uniquely minimizes the Laplacian spectral radius for n ⩾ 2g − 1 when g is even and for n ⩾ 3g − 1 when g is odd.
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